Quasi-2D Coulomb systems are of fundamental importance and have attracted much attention in many areas nowadays. Their reduced symmetry gives rise to interesting collective behaviors, but also brings great challenges for particle-based simulations. Here, we propose a novel algorithm framework to address the $O(N^2)$ simulation complexity associated with the long-range nature of Coulomb interactions. First, we introduce an efficient Sum-of-Exponentials (SOE) approximation for the long-range kernel associated with Ewald splitting, achieving uniform convergence in terms of inter-particle distance, which reduces the complexity to $O(N^{7/5})$. We then introduce a random batch sampling method in the periodic dimensions, the stochastic approximation is proven to be both unbiased and with reduced variance via a tailored importance sampling strategy, further reducing the computational cost to $O(N)$. The performance of our algorithm is demonstrated via various numerical examples. Notably, it achieves a speedup of $2\sim 3$ orders of magnitude comparing with Ewald2D method, enabling molecular dynamics (MD) simulations with up to $10^6$ particles on a single core. The present approach is therefore well-suited for large-scale particle-based simulations of Coulomb systems under confinement, making it possible to investigate the role of Coulomb interaction in many practical situations.

翻译：准二维库仑系统具有基础重要性，并在当今诸多领域备受关注。其对称性的降低引发了有趣的集体行为，但也给基于粒子的模拟带来了巨大挑战。本文提出了一种新颖的算法框架，以应对由库仑相互作用长程特性所导致的 $O(N^2)$ 模拟复杂度。首先，我们针对与Ewald分裂相关的长程核函数，引入了一种高效的指数和（SOE）近似，该近似在粒子间距离方面实现了均匀收敛，从而将复杂度降低至 $O(N^{7/5})$。随后，我们在周期性维度中引入了随机批采样方法，通过定制的重要性采样策略，证明了该随机近似既是无偏的，又具有降低的方差，从而进一步将计算成本降至 $O(N)$。我们通过多个数值算例展示了该算法的性能。值得注意的是，与Ewald2D方法相比，其实现了 $2\sim 3$ 个数量级的加速，使得在单核上能够进行高达 $10^6$ 个粒子的分子动力学（MD）模拟。因此，本方法非常适用于受限条件下库仑系统的大规模粒子模拟，使得研究库仑相互作用在许多实际场景中的作用成为可能。